There has recently been a considerable amount of activity in developing adaptive methods for the selection of primal constraints for BDDC algorithms and, in particular, for BDDC deluxe variants.

Much of the earlier work for adaptive BDDC and FETI-DP iterative substructuring algorithms, which has been supported by theory, has been confined to developing primal constraints for equivalence classes related to two subdomain boundaries such as those for the subdomain edges for problems defined on domains in the plane; see, in particular, the paper by Klawonn, Radtke, and Rheinbach [14].

resulting in a linear system of equations to be solved using BDDC domain decomposition algorithms, in particular, its deluxe variant.

Scalar elliptic problems in the plane are analyzed in [7, 9]; [17] includes a FETI-DP algorithm for scalar elliptic and elasticity problems, and [5, 10] include an iterative substructuring method and a BDDC deluxe algorithm for problems in H(curl) in 2D, respectively.

In addition, a BDDC algorithm with deluxe scaling is considered in [5] for uniform domains in 2D, and in [11] for 3D.

Recently, in [11], new tools are developed for more general subdomains and a BDDC deluxe method, where the faces are assumed to be only star-shaped polygons.

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